On the similarity principle for planar vector fields: Application to second order PDE
The author is mainly interested in the local solvability of the equation (1) \(Lw=Aw=B\overline w\) where \(L\) is a nonsingular, integrable, \(\mathbb{C}\)-valued vector field in the plane, and \(A\) and \(B\) are smooth functions. As an application, the solutions of the second order partial differential equation (2) \(a\varphi_{xx}+ 2b\varphi_{xy} +c\varphi_{yy} +\alpha\varphi_x + \delta \varphi_y= 0\) which is elliptic everywhere except along the curve \(\Sigma= \{(x,y)\mid \rho^2(x,y)= b^2(x,y)-a(x,y) \cdot c(x,y)=0\}\) for which \(\Sigma\) is a characteristic curve, are expressed in terms of the solutions of an equation of type (1). To this end the author investigates in the paper an extension to which similarity principle holds for nonelliptic vector fields. As a paradigm to vector fields the author considers three typical vector fields: \(L_1 ={\partial \over \partial y}-3iy^2{\partial\over\partial x}\), \(L_2= {\partial \over\partial y}-i x{\partial\over \partial x}\), and \(M={\partial\over \partial y}-2iy{\partial \over\partial x}\).
scientific article; zbMATH DE number 1346389
scientific article